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The impact of load placement on grid resonances during grid restoration

Adolfo Anta, Diego Cifelli

TL;DR

The paper addresses damping of grid resonances during backbone energization with inverter-based generation. It develops a spectral analysis framework by modeling a transmission-line backbone with a load and exploiting the 2-Toeplitz structure and Chebyshev polynomials to obtain analytical expressions for the eigenvalues as a function of load position. The key finding is that placing the load at the line center maximizes damping of the first resonance, though no single location eliminates all resonances; distributed loading along the line provides more robust damping for multiple resonances. The results yield practical guidelines for grid restoration operations and highlight limitations when only a single load can be connected, motivating extension to more complex topologies and graph-dependent effects.

Abstract

As inverter-based generation is being massively deployed in the grid, these type of units have to take over the current roles of conventional generation, including the capability of restoring the grid. In this context, the resonances of the grid during the first steps of a black start can be concerning, given that the grid is lightly loaded. Especially relevant are the low frequency resonances, that may be excited by the harmonic components of the inverter. A typical strategy to avoid or minimize the effect of such resonances relies on connecting load banks. This was fairly feasible with conventional generation, but given the limited ratings of inverters, the amount of load that can be connected at the beginning is very limited. In this paper we consider the energization of a transmission line, and investigate the optimal location of a load along a line in order to maximize the damping in the system. By analysing the spectral properties as a function of the load location, we formally prove that placing the load in the middle of the transmission line maximizes the damping ratio of the first resonance of the system.

The impact of load placement on grid resonances during grid restoration

TL;DR

The paper addresses damping of grid resonances during backbone energization with inverter-based generation. It develops a spectral analysis framework by modeling a transmission-line backbone with a load and exploiting the 2-Toeplitz structure and Chebyshev polynomials to obtain analytical expressions for the eigenvalues as a function of load position. The key finding is that placing the load at the line center maximizes damping of the first resonance, though no single location eliminates all resonances; distributed loading along the line provides more robust damping for multiple resonances. The results yield practical guidelines for grid restoration operations and highlight limitations when only a single load can be connected, motivating extension to more complex topologies and graph-dependent effects.

Abstract

As inverter-based generation is being massively deployed in the grid, these type of units have to take over the current roles of conventional generation, including the capability of restoring the grid. In this context, the resonances of the grid during the first steps of a black start can be concerning, given that the grid is lightly loaded. Especially relevant are the low frequency resonances, that may be excited by the harmonic components of the inverter. A typical strategy to avoid or minimize the effect of such resonances relies on connecting load banks. This was fairly feasible with conventional generation, but given the limited ratings of inverters, the amount of load that can be connected at the beginning is very limited. In this paper we consider the energization of a transmission line, and investigate the optimal location of a load along a line in order to maximize the damping in the system. By analysing the spectral properties as a function of the load location, we formally prove that placing the load in the middle of the transmission line maximizes the damping ratio of the first resonance of the system.
Paper Structure (8 sections, 18 equations, 4 figures)

This paper contains 8 sections, 18 equations, 4 figures.

Table of Contents

  1. Introduction and motivation
  2. Modelling
  3. Spectral analysis and analytical characterization
  4. An overview of 2-Toeplitz matrices and Chebyshev polynomials
  5. The unloaded case
  6. The effect of the load location
  7. An academic example
  8. Conclusions and Outlook

Figures (4)

  • Figure 1: Transmission line model with cascaded $\pi$ networks and a load connected at section $z$.
  • Figure 2: Damping factor of the first resonance as the load is connected along the transmission line.
  • Figure 3: Damping factor of the higher resonances as the load is connected along the transmission line.
  • Figure 4: Root locus with a load located at the center point.